A Legendre polynomial solver for the Langevin Boltzmann equation

Due to the proliferation of wireless and other RF applications, noise simulation has become an important topic of TCAD. Although the so-called physical Monte Carlo (MC) method inherently contains electronic noise, this time-domain based method is far too slow for most noise calculations, which are performed in the GHz range or below, because the CPU time is at least inversely proportional to the minimum frequency investigated. On the other hand, the Langevin Boltzmann equation (LBE), which is the basis of the MC method, can be also solved directly in the frequency domain by other numerical methods, thus avoiding the CPU time increase at low frequencies. Demonstrated in this paper is the first numerical solver for the LBE in the frequency domain, which was successfully verified by comparison with MC results. It was found that noise calculation requires a Legendre polynomial expansion up to the third order. Nevertheless, the new method is orders of magnitude faster than corresponding MC simulations.